Free materials · Manchester & St Andrews
Econometrics lecture notes,
exercises & solutions
Free lecture slides, written notes, problem sets and fully worked solutions from four university econometrics courses taught by Dr Nicky Grant. All materials are the actual files used in the courses, not simplified summaries. No registration required.
ECON30401
University of Manchester · 3rd year undergraduate · Dr Grant taught Lectures 1–4
Univariate time series modelling from first principles. Nicky taught the first half of this course; Lectures 5–8 (VAR systems, unit roots, cointegration) were taught by Prof. Alastair R. Hall and are not included here.
Lecture slides
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Lecture 1 · 2019 version (latest)
Introduction to Time Series Econometrics
Time series process vs. realisation. Strict and covariance stationarity formally defined. White noise. MA(q) process, definition, mean, variance, autocovariance function. Sample realisations of AR, MA and ARMA processes. Sets up all notation used in subsequent lectures.
Lecture 2 · 2016 version (latest)
ARMA(p,q) Processes & Stationarity
MA(∞) process and squared/absolute summability. Wold Decomposition Theorem. AR(1) as MA(∞), mean α/(1−φ₁), variance σ²/(1−φ₁²), ACF ρ(k)=φ₁ᵏ. ARMA(1,1) properties. Lag operator introduced. General ARMA(p,q). Stationarity via characteristic polynomial, roots of 1−φ₁L−…−φₚLᵖ must lie outside the unit circle.
Lecture 3 · 2015 version
Estimation, Hypothesis Testing & Model Selection
Sample ACF: ρ̂(k) = Σ(Yₜ−Ȳ)(Yₜ₋ₖ−Ȳ)/Σ(Yₜ−Ȳ)². Under H₀: ρ(k)=0, √T·ρ̂(k) →ᵈ N(0,1). Correlogram and 95% bands ±1.96/√T. Ljung-Box portmanteau test Q(m)~χ²(m). AR estimation by OLS; MA and ARMA by MLE. AIC = −2·log(L̂)+2K; BIC = −2·log(L̂)+K·log(T). Applied to UK GDP growth data.
Lecture 4 · 2015 version
Prediction & Seasonality
AR(1) one-step predictor Ŷₜ₊₁=µ+φ₁yₜ. J-step ahead prediction and forecast error variance σ²·Σφ₁²⁽ᴶ⁻ʲ⁾. Long-horizon convergence to unconditional mean. Deterministic seasonality: seasonal dummies, dummy variable trap, seasonal adjustment by OLS residuals. Stochastic seasonality: SAR(1), its MA(∞) form, ACF. Combined model with dummies and AR dynamics. UK retail sales application.
Written notes
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Manchester
Weeks 1–2 · Comprehensive notes (final version)
Time Series Econometrics. Lecture Notes Weeks 1 & 2
Written prose notes (not slides) covering the full material from Lectures 1–3. Full derivations of MA(q) and AR(1) properties, Wold decomposition proof, ACF and PACF theory, sample ACF distribution under the null, model estimation rationale, information criteria derivation. Suitable for reading alongside or instead of slides. Best reference for exam revision.
Foundation · OLS and linear models
OLS Notes (University of Manchester, hosted externally)
The linear model assumed throughout both ECON30401 and ECON61001. Covers OLS estimator, Gauss-Markov theorem, large-sample properties, and inference. Hosted publicly by the University of Manchester.
Exercises & solutions
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Exercise 1 · ARMA properties
Problem Set 1: ARMA Properties & Wold Decomposition
Deriving MA(∞) representations, showing AR(1) properties from MA(∞) form, ARMA(1,1) properties, stationarity conditions. Video solutions provided separately.
Video Exercise 2 · AR(2) and ARMA(1,1)
Video Exercise 2: AR(2) Prediction & ARMA(1,1) Moments
Deriving 1, 2 and 3-step ahead forecasts and forecast error variances for AR(2). Deriving mean, variance and ACF for ARMA(1,1) via the MA(∞) route. Supported by video recordings.
Class Exercise 2 · IBM stock returns
Applied Exercise: ARMA Modelling of IBM Monthly Returns
IBM monthly stock returns 1991–2013 (T=276). Correlogram examination, white noise testing, model identification, estimation, diagnostics. Requires interpreting EViews output.
PC Lab 1 · EViews, UK real GDP
PC Lab 1: Time Series in EViews. UK Real GDP 1955–2013
Hands-on EViews exercise. UK quarterly real GDP (seasonally adjusted). Plot, growth rates, ACF/PACF, ARMA estimation, information criteria for model selection, forecasting, residual diagnostics.
Lectures 5–8 of ECON30401 (VAR systems, unit roots, cointegration) were taught by Prof. Alastair R. Hall and are not available here.
ECON61001
University of Manchester · Postgraduate MSc · Dr Grant taught Lectures 5–7
Diagnostic testing, endogeneity and instrumental variables, the three topics most relevant to applied econometric research. These lectures cover material central to any serious empirical economics paper.
Lecture slides
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Lecture 5 · Diagnostic testing
Diagnostic Testing & Robust Inference
Testing OLS assumptions in practice. Heteroskedasticity: White test (T·R² ~χ²), Breusch-Pagan; heteroskedasticity-consistent (HC) standard errors. Serial correlation: Durbin-Watson, Breusch-Godfrey LM test; HAC (Newey-West) standard errors. Simulation evidence on test size across sample sizes. 2016 version.
Lecture 6 · Instrumental variables
Endogeneity & Instrumental Variables (2SLS)
When E[uᵢxᵢ]≠0. OLS biased and inconsistent. Sources: simultaneity, omitted variables, measurement error. IV estimator. 2SLS: first stage regresses xᵢ on zᵢ; second stage uses x̂ᵢ. Validity conditions: relevance (Cov(zᵢ,xᵢ)≠0) and exclusion restriction (Cov(zᵢ,uᵢ)=0). First-stage F>10 rule of thumb. Acemoglu-Johnson-Robinson settler mortality example. 2016 version.
Lecture 7 · Weak instruments
Instrumental Variables with Weak / Unidentified Instruments
Weak instrument problem: 2SLS severely biased toward OLS; standard Wald confidence intervals have severe size distortions. Concentration parameter. Stock-Yogo (2005) critical values (F>16.38 for 5% maximal size distortion with one instrument). Anderson-Rubin identification-robust test. LIML as less biased alternative to 2SLS. Conditional likelihood ratio (CLR) test. Directly connected to Dr Grant's CeMMAP research on singular moment variance.
Tutorial problem sets & solutions
Tutorial 3–4 · WLS and GLS
Tutorials 3 & 4: Weighted Least Squares & Generalised Least Squares
Deriving the WLS estimator, properties under heteroskedasticity, GLS as efficient estimator when errors are non-spherical. Problems and full worked solutions included.
Tutorial 5 · Robust inference
Tutorial 5: Heteroskedasticity & Serial Correlation. Applied Problems
White test for heteroskedasticity, linear-form test, HC standard errors in practice. Serial correlation testing, Newey-West HAC standard errors. Interpretation of regression output with diagnostics. Full solutions included.
Lectures 1–4 of ECON61001 (OLS foundations, large-sample theory, hypothesis testing) were taught by other faculty. The OLS Notes linked above cover that material.
EC5221
University of St Andrews · Postgraduate MSc · Semester 2 · Full course
Postgraduate-level time series covering the same core theory as ECON30401 but with greater mathematical rigour, formal asymptotic arguments, and extension to VAR systems. Latest versions are from 2020/2021.
Lecture slides
Week 1 · 2021 version (latest)
Stationary Univariate Time Series. ARMA Processes & Stationarity
Covers the full formal treatment: time series process vs. realisation, strict and covariance stationarity, bivariate moments and the autocovariance function, white noise, ARMA(p,q), moments and stationarity conditions via the lag operator, MA(∞) and Wold decomposition, AR(1) as MA(∞). More formal and complete than the ECON30401 version of the same material.
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Week 2 · 2021 version (latest)
Stationary Univariate Time Series. Estimation & Inference
Sample moments: sample mean, sample ACF and their asymptotic distributions. Statistical properties of sample ACF under H₀ and H₁. Portmanteau tests. Maximum likelihood estimation of ARMA processes: Gaussian log-likelihood, conditional vs. exact MLE, Newton-Raphson. AIC and BIC for order selection. Reading list and applications.
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6 MB
Problem sets & solutions
Solutions
Problem Set 2 · VAR systems · 2021 version (latest)
Problem Set 2: Bivariate VAR(1) Systems
Bivariate VAR(1): Yₜ = α + Φ₁Yₜ₋₁ + εₜ. Stationarity condition via eigenvalues of Φ₁. Mean, variance and autocovariance of stationary VAR(1). Impulse response functions. Granger causality testing. With specific parameter values (φ₁₁=0.6, φ₁₂=0.1, φ₂₁=−0.6, φ₂₂=1.1).
Video Problem Set 1 · Solutions · 2021 version (latest)
Video Exercise Sheet 1 Solutions: MA(∞) and ARMA(1,1)
Full worked solutions to the first video problem set. Deriving that a given MA(∞) process with recursively defined coefficients (η₀=1, η₁=φ₁+θ₁, ηₛ=φ₁ηₛ₋₁ for s≥2) is an ARMA(1,1). Mean, variance, and ACF derivation. Stationarity and invertibility discussion.
EC5609
University of St Andrews · Postgraduate MSc · Semester 1 · Full course
The statistical properties of financial data, ARMA modelling of returns, maximum likelihood, and advanced volatility modelling with GARCH. Applied throughout with stock return data.
Lecture slides
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Lecture 1 · 2021/22 version (latest)
Distribution Theory & Properties of Stock Price Returns
Financial econometric data. Log returns rₜ=log(Pₜ)−log(Pₜ₋₁). Statistical distribution theory: PDF, CDF, moments. Sample mean, variance, skewness (third standardised central moment), kurtosis (fourth). Leptokurtosis in returns: excess kurtosis>0, fat tails. Normal vs. Student-t. Central Limit Theorem and asymptotic normality of sample moments. Sampling distributions. Motivation for ARMA and GARCH modelling of returns.
Lecture 7 · 2021/22 version (latest)
Advanced Volatility Modelling. ARCH & GARCH
Conditional heteroskedasticity in financial returns. ARCH(q): hₜ=ω+Σαᵢεₜ₋ᵢ². GARCH(1,1): hₜ=ω+α·εₜ₋₁²+β·hₜ₋₁; stationarity requires α+β<1. Volatility clustering. Asymmetric effects (leverage): EGARCH (Nelson) log(hₜ)=ω+α·|εₜ₋₁/√hₜ₋₁|+γ·εₜ₋₁/√hₜ₋₁+β·log(hₜ₋₁). TARCH/GJR-GARCH. GARCH-in-mean. Generalised error distribution for fat tails. MLE of GARCH models. Empirical examples.
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11 MB
Maximum likelihood · supplementary notes
Maximum Likelihood Estimation in Financial Econometrics
MLE as a general estimator for non-linear models. Requires assumption about the distribution of data. Log-likelihood for normal returns: L=−T/2·log(2πσ²)−Σ(yᵢ−µ)²/(2σ²). Conditional log-likelihood for GARCH with time-varying hₜ. Broad conceptual treatment for MSc level.
Problem sets & solutions
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Problem Set 1 · 2023/24 version (latest)
Problem Set 1: Applied. US Stock Market Returns (MktRF)
Applied problems using Stata output for MktRF, weekly excess US stock market returns (%) March 2000–May 2021 (T=255). Statistical properties, ARMA modelling, diagnostics. Most recent version of the course problem set.
Problem Set 1 · 2021/22 version + solutions
Problem Set 1: Distribution Theory. Uniform, Normal, Student-t
Theoretical problems. Uniform X~Unif[0,1]: mean, variance, skewness, kurtosis. AR(1) and MA(1) properties review. Moment estimation from financial data. Full worked solutions available.
Problem Set 2 · ARMA processes review
Problem Set 2: MA(1), AR(1) Processes & Moment Derivation
MA(1) process definition, mean, variance, ACF. AR(1) process, mean, variance, ACF, stationarity. Bridges between the time series and financial econometrics material.
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Related resources
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Time series guide, unit roots, cointegration, ARIMA
Plain-English guide to the topics in Lectures 5–8 of ECON30401 (not included above). Unit root tests, ARIMA, GARCH, cointegration.
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What is OLS? The foundation for all material above
All four courses assume knowledge of OLS. This guide explains the estimator, Gauss-Markov theorem, and inference from first principles.
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YouTube lectures, 30+ applied econometrics videos
The @economaths channel covers applied econometrics across all levels. Free to watch. Complements the lecture notes above.
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Ask a question, forums and free help
r/econometrics, Cross Validated, TheStudentRoom, Economics Stack Exchange, where to get free help, and where Dr Grant answers questions.
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